Winning percentages

The basis of an Elo system is estimating the chances that one player will defeat another based on their ratings. The ratings used by the EGF depend in part upon your rank, with 20k-1k ranks distributed over the range 100-2000, 1d-7d distributed over the range 2100-2700 and pro ranks above that. The rating that (nominally) belongs to a certain rank can be seen in the table below.

The chance Se(A) of player A (with the lower rating) winning against a higher rated opponent is given by the following formula:

The chance that player B wins is equal to the chance that A does not win, and can easily be derived from the table below by subtracting the percentages from 100%.

where D is the rating difference and a is a factor that depends on rating, and varies from a value of 200 at 20 kyu to a value of 70 at 7 dan. The table below lists a and shows the estimated winning percentage according to the EGF formula. The same data has been used to generate the graph to the right.

As we can see, the graph gets steeper for higher rated players. What this means is that the EGF rating considers it more likely for a 5 dan to win against a 4 dan, than for a 4 kyu to win against a 5 kyu, but also more likely for the 5 dan to lose to a 6 dan than for the 4 kyu to lose to a 3 kyu. This represents the fact that stronger players tend to have a more stable performance, with fewer "upset" victories or defeats.

Estimated winning percentage against a player X rating points stronger, where X is:

rating

rank

K

a

0

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

100

20k

116

200

50.0%

46.9%

43.8%

40.7%

37.8%

34.9%

32.1%

29.4%

26.9%

24.5%

22.3%

20.2%

18.2%

16.5%

14.8%

13.3%

11.9%

10.7%

9.5%

8.5%

7.6%

200

19k

110

195

50.0%

46.8%

43.6%

40.5%

37.5%

34.5%

31.7%

29.0%

26.4%

24.0%

21.7%

19.6%

17.7%

15.9%

14.2%

12.8%

11.4%

10.2%

9.0%

8.0%

7.1%

300

18k

105

190

50.0%

46.7%

43.5%

40.3%

37.1%

34.1%

31.2%

28.5%

25.9%

23.4%

21.2%

19.0%

17.1%

15.3%

13.7%

12.2%

10.9%

9.6%

8.6%

7.6%

6.7%

400

17k

100

185

50.0%

46.6%

43.3%

40.0%

36.8%

33.7%

30.8%

28.0%

25.3%

22.9%

20.6%

18.4%

16.5%

14.7%

13.1%

11.6%

10.3%

9.1%

8.1%

7.1%

6.3%

500

16k

95

180

50.0%

46.5%

43.1%

39.7%

36.5%

33.3%

30.3%

27.4%

24.8%

22.3%

20.0%

17.8%

15.9%

14.1%

12.5%

11.1%

9.8%

8.6%

7.6%

6.7%

5.9%

600

15k

90

175

50.0%

46.4%

42.9%

39.4%

36.1%

32.9%

29.8%

26.9%

24.2%

21.7%

19.3%

17.2%

15.3%

13.5%

11.9%

10.5%

9.2%

8.1%

7.1%

6.2%

5.4%

700

14k

85

170

50.0%

46.3%

42.7%

39.1%

35.7%

32.4%

29.3%

26.3%

23.6%

21.0%

18.7%

16.6%

14.6%

12.9%

11.3%

9.9%

8.7%

7.6%

6.6%

5.8%

5.0%

800

13k

80

165

50.0%

46.2%

42.5%

38.8%

35.3%

31.9%

28.7%

25.7%

22.9%

20.4%

18.0%

15.9%

14.0%

12.2%

10.7%

9.3%

8.1%

7.1%

6.1%

5.3%

4.6%

900

12k

75

160

50.0%

46.1%

42.3%

38.5%

34.9%

31.4%

28.1%

25.1%

22.3%

19.7%

17.3%

15.2%

13.3%

11.6%

10.1%

8.8%

7.6%

6.6%

5.7%

4.9%

4.2%

1000

11k

70

155

50.0%

46.0%

42.0%

38.1%

34.4%

30.9%

27.5%

24.4%

21.6%

19.0%

16.6%

14.5%

12.6%

10.9%

9.5%

8.2%

7.0%

6.1%

5.2%

4.5%

3.8%

1100

10k

65

150

50.0%

45.8%

41.7%

37.8%

33.9%

30.3%

26.9%

23.7%

20.9%

18.2%

15.9%

13.8%

11.9%

10.3%

8.8%

7.6%

6.5%

5.6%

4.7%

4.0%

3.4%

1200

9k

60

145

50.0%

45.7%

41.5%

37.3%

33.4%

29.7%

26.2%

23.0%

20.1%

17.5%

15.1%

13.0%

11.2%

9.6%

8.2%

7.0%

6.0%

5.1%

4.3%

3.6%

3.1%

1300

8k

55

140

50.0%

45.5%

41.2%

36.9%

32.9%

29.1%

25.5%

22.3%

19.3%

16.7%

14.4%

12.3%

10.5%

8.9%

7.6%

6.4%

5.4%

4.6%

3.9%

3.3%

2.7%

1400

7k

51

135

50.0%

45.4%

40.8%

36.5%

32.3%

28.4%

24.8%

21.5%

18.5%

15.9%

13.6%

11.5%

9.8%

8.3%

7.0%

5.9%

4.9%

4.1%

3.4%

2.9%

2.4%

1500

6k

47

130

50.0%

45.2%

40.5%

36.0%

31.7%

27.7%

24.0%

20.6%

17.7%

15.0%

12.8%

10.8%

9.0%

7.6%

6.3%

5.3%

4.4%

3.7%

3.0%

2.5%

2.1%

1600

5k

43

125

50.0%

45.0%

40.1%

35.4%

31.0%

26.9%

23.1%

19.8%

16.8%

14.2%

11.9%

10.0%

8.3%

6.9%

5.7%

4.7%

3.9%

3.2%

2.7%

2.2%

1.8%

1700

4k

39

120

50.0%

44.8%

39.7%

34.9%

30.3%

26.1%

22.3%

18.9%

15.9%

13.3%

11.1%

9.2%

7.6%

6.2%

5.1%

4.2%

3.4%

2.8%

2.3%

1.9%

1.5%

1800

3k

35

115

50.0%

44.6%

39.3%

34.2%

29.5%

25.2%

21.3%

17.9%

14.9%

12.4%

10.2%

8.4%

6.9%

5.6%

4.5%

3.7%

3.0%

2.4%

2.0%

1.6%

1.3%

1900

2k

31

110

50.0%

44.3%

38.8%

33.6%

28.7%

24.3%

20.4%

16.9%

14.0%

11.5%

9.3%

7.6%

6.1%

5.0%

4.0%

3.2%

2.6%

2.1%

1.6%

1.3%

1.1%

2000

1k

27

105

50.0%

44.1%

38.3%

32.9%

27.8%

23.3%

19.3%

15.9%

13.0%

10.5%

8.5%

6.8%

5.4%

4.3%

3.4%

2.7%

2.2%

1.7%

1.4%

1.1%

0.8%

2100

1d

24

100

50.0%

43.8%

37.8%

32.1%

26.9%

22.3%

18.2%

14.8%

11.9%

9.5%

7.6%

6.0%

4.7%

3.7%

2.9%

2.3%

1.8%

1.4%

1.1%

0.9%

0.7%

2200

2d

21

95

50.0%

43.5%

37.1%

31.2%

25.9%

21.2%

17.1%

13.7%

10.9%

8.6%

6.7%

5.2%

4.1%

3.2%

2.5%

1.9%

1.5%

1.1%

0.9%

0.7%

0.5%

2300

3d

18

90

50.0%

43.1%

36.5%

30.3%

24.8%

20.0%

15.9%

12.5%

9.8%

7.6%

5.9%

4.5%

3.4%

2.6%

2.0%

1.5%

1.2%

0.9%

0.7%

0.5%

0.4%

2400

4d

15

85

50.0%

42.7%

35.7%

29.3%

23.6%

18.7%

14.6%

11.3%

8.7%

6.6%

5.0%

3.8%

2.8%

2.1%

1.6%

1.2%

0.9%

0.7%

0.5%

0.4%

0.3%

2500

5d

13

80

50.0%

42.3%

34.9%

28.1%

22.3%

17.3%

13.3%

10.1%

7.6%

5.7%

4.2%

3.1%

2.3%

1.7%

1.2%

0.9%

0.7%

0.5%

0.4%

0.3%

0.2%

2600

6d

11

75

50.0%

41.7%

33.9%

26.9%

20.9%

15.9%

11.9%

8.8%

6.5%

4.7%

3.4%

2.5%

1.8%

1.3%

0.9%

0.7%

0.5%

0.3%

0.2%

0.2%

0.1%

2700

7d

10

70

50.0%

41.2%

32.9%

25.5%

19.3%

14.4%

10.5%

7.6%

5.4%

3.9%

2.7%

1.9%

1.4%

1.0%

0.7%

0.5%

0.3%

0.2%

0.2%

0.1%

0.1%

For comparison, these are the winning percentages in the basic Elo formula used in Chess (percentages not rating dependent). This is equivalent to using a value of 173.7 for the variable 'a' in the formula above.:

Distribution of GoR per rank

Although there is a nominal rating associated with each rank, the actual rating of players of the same rank varies. The graph below shows how it varies, with most ranks roughly showing a gaussian distribution around the nominal rating:

Method of generating the graph

This graph plots the frequency of post-tournament ratings per grade in the EGF rating database, for all tournaments in or after the year 2000. The height of the graph at rating point X represents how often a certain post tournament rating occurred.

To smooth the graph, each data point also affects its neighbours, adjusted by distance. A rating at 0 points distance adds 25, at 1 point distance 24, at 5 points distance 20, and so on downto 0 at 25 points distance or more. Since this method adds 625 to the volume of each rating graph for each data point, the end result was scaled back by that factor.

I have discarded any data points where a pre tournament rating was reset based on a jump of at least 2 grades, and where the post tournament rating was 100 points lower than that reset grade. Because a player can never lose more than 100 points in one tournament, such data point are very common among lower kyu
players (because ratings change more quickly the lower they are). This caused very noticable artificial bumps around those ratings.

The graph does not plot values below value 25 to eliminate artifacts caused by single outliers (there are some errors in the EGF data, I have informed Ales Cieply of those I have found).

7D includes all Pro grades, which results in noticable bumps, especially around 2760 (3p) and 2820 (5p).